Global Positioning System Reference
In-Depth Information
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can be substituted into Equations (2.99) to (2.101) to obtain the topocentric geode-
tic azimuth, elevation, and distance of the satellite. The geodetic latitude and longi-
tude in these expressions can be computed from x P , if necessary. For low-accuracy
applications such as the creation of visibility charts it is sufficient to use spherical
approximations.
3. 1.4 Perturbed Satellite Motion
The accurate determination of satellite positions must consider various disturbing
forces. Disturbing forces are all those forces causing the satellite to deviate from the
simple normal orbit. The disturbances are caused primarily by the nonsphericity of the
gravitational potential, the attraction of the sun and the moon, the solar radiation pres-
sure, and other smaller forces acting on the satellites. For example, albedo is a force
due to electromagnetic radiation reflected by the earth. There could be thermal rera-
diation forces caused by anisotropic radiation from the surface of the spacecraft. Ad-
ditional forces, such as residual atmospheric drag, affect satellites closer to the earth.
Several of the disturbing forces can be readily computed; others, in particular the
smaller forces, require detailed modeling and are still subject to further research.
Knowing the accurate location of the satellites, i.e., being able to treat satellite po-
sition coordinates as known quantities, is important in surveying, in particular for
long baseline determination. Most scientific applications of GPS demand the highest
orbital accuracy, all the way down to the centimeter level. However, even surveying
benefits from such accurate orbits, e.g., in precise point positioning with one receiver.
See Section 7.5 for additional detail on this technique. One of the goals of the Inter-
national GPS Service (IGS) and its contributing agencies and research groups is to
refine continuously orbital computation and modeling and to make the most accurate
satellite ephemeris available to the users. In this section, we provide only an intro-
ductory exposition of orbital determination. The details are found in the extensive
literature, going all the way back to the days of the first artificial satellites.
The equations of motion are expressed in an inertial (celestial) coordinate system,
corresponding to the epoch of the initial conditions. The initial conditions are either
( X , X ) or the Kepler elements at a specified epoch. Because of the disturbing forces,
all Kepler elements are functions of time. The transformation given above can be used
to transform the initial conditions from ( X , X ) to Kepler elements and vice versa. The
equations of motion, as expressed in Cartesian coordinates, are
[66
Lin
3.2
——
Nor
PgE
[66
d X
dt
X
=
(3.68)
d X
dt
=− µ
X
X g +
X s +
X m +
X SRP +···
3 +
(3.69)
X
These are six first-order differential equations. The symbol
denotes the geocentric
gravitational constant (3.5). The first term in (3.69) represents the acceleration of the
central gravity field that generates the normal orbits discussed in the previous section.
µ
 
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